The Scottish champions will meet Swedish title holders Elfsborg in the first leg of their third round qualifier after completing the formalities of their win over the Northern Irish part-timers having strolled to a 3-0 win at Solitude a week ago.
Elfsborg won 4-0 against Daugava in Latvia earlier in the day to progress 11-1 over the two legs.
Celtic were wasteful against Cliftonville, but managed to take two of the numerous chances that came their way as Ambrose nodded Anthony Stokes's cross into the net for the opening goal with Samaras adding the second when he rose to bury a head from Kris Commons's corner-kick.
Celtic brought Gary Hooper - linked with a £6 million move to QPR - on for the closing 20 minutes, and he almost added the third, but was unfortunate to see his effort come back off a post.
Ambrose, who had cut short his delayed summer break for last week's first leg in Belfast when manager Neil Lennon found himself short of defenders, put Celtic ahead in the 16th minute when he headed in an Anthony Stokes cross after clever footwork from Mikael Lustig on the right.
As the procession towards the Reds goal continued in the first half James Forrest skimmed Conor Devlin's bar with a right-footed drive in the 38th minute.
However, there was no addition to the score until the 70th minute when Samaras, who had scored last week in Belfast, made it 2-0 when he headed in a Kris Commons cross from close range.
Hooper replaced Beram Kayal and hit the outside of the post with an overhead kick but the visitors held out to leave Glasgow with a respectable scoreline.
Elfsborg, well into their league season, will provide a tougher for Celtic test than Cliftonville whether or not Mo Bangura, on loan from the Hoops, plays against his parent club and that is the issue which will surround the tie until kick-off time at Parkhead next week.
Second qualifying round results:
FH (Iceland) - Ekranas Panevezys (Lithuania) 2-1 (halftime: 1-1)
First leg: Ekranas Panevezys - FH 0-1. FH win 3-1 on aggregate.
Zeljeznicar Sarajevo (Bosnia & Herzegovina) - Viktoria Plzen (Czech Republic) 1-2 (halftime: 1-2)
First leg: Viktoria Plzen - Zeljeznicar Sarajevo 4-3. Viktoria Plzen win 6-4 on aggregate.
Celtic (Scotland) - Cliftonville (Northern Ireland) 2-0 (halftime: 1-0)
First leg: Cliftonville - Celtic 0-3. Celtic win 5-0 on aggregate.
Dinamo Zagreb (Croatia) - CS Fola Esch (Luxembourg) 1-0 (halftime: 0-0)
First leg: CS Fola Esch - Dinamo Zagreb 0-5. Dinamo Zagreb win 6-0 on aggregate.
Vardar Skopje (Macedonia FYR) - Steaua Bucharest (Romania) 1-2 (halftime: 1-1)
First leg: Steaua Bucharest - Vardar Skopje 3-0. Steaua Bucharest win 5-1 on aggregate.
Maccabi Tel Aviv (Israel) - Gyor ETO (Hungary) 2-1 (halftime: 1-0)
First leg: Gyor ETO - Maccabi Tel Aviv 0-2. Maccabi Tel Aviv win 4-1 on aggregate.
EB/Streymur (Faroe Islands) - Dynamo Tbilisi (Georgia) 1-3 (halftime: 1-2)
First leg: Dynamo Tbilisi - EB/Streymur 6-1. Dynamo Tbilisi win 9-2 on aggregate.
Skenderbeu (Albania) - Neftchi Baku (Azerbaijan) 1-0 (halftime: 0-0, 90 mins: 0-0) AET
First leg: Neftchi Baku - Skenderbeu 0-0. Skenderbeu win 1-0 on aggregate.
Molde (Norway) - Sligo Rovers (Ireland) 2-0 (halftime: 1-0)
First leg: Sligo Rovers - Molde 0-1. Molde win 3-0 on aggregate.
Kalju Nomme (Estonia) - HJK Helsinki (Finland) 2-1 (halftime: 2-0)
First leg: HJK Helsinki - Kalju Nomme 0-0. Kalju Nomme win 2-1 on aggregate.
Sutjeska Niksic (Montenegro) - Sheriff Tiraspol (Moldova) 0-5 (halftime: 0-1)
First leg: Sheriff Tiraspol - Sutjeska Niksic 1-1. Sheriff Tiraspol win 6-1 on aggregate.
Daugava Daugavpils (Latvia) - Elfsborg Boras (Sweden) 0-4 (halftime: 0-1)
First leg: Elfsborg Boras - Daugava Daugavpils 7-1. Elfsborg Boras win 11-1 on aggregate.
Shakhtyor IK (Kazakhstan) - BATE Borisov (Belarus) 1-0 (halftime: 0-0)
First leg: BATE Borisov - Shakhtyor IK 0-1. Shakhtyor IK win 2-0 on aggregate.
Wednesday, July 24
Ludogorets (Bulgaria) v Slovan Bratislava (Slovakia)
Maribor (Slovenia) v Birkirkara (Malta)
Partizan Belgrade (Serbia) v Shirak Gyumri (Armenia)
Legia Warsaw (Poland) v The New Saints (Wales)